Discrete Curvature and Applications in Graph-Based Learning

The problem of identifying geometric structure in heterogeneous, high-dimensional data is a cornerstone of Representation Learning. In this talk, we study the problem of data geometry from the perspective of Discrete Geometry. We start by reviewing discrete notions of curvature with a focus on discrete Ricci curvature. Then we discuss how curvature is linked to mesoscale structure in graphs, which gives rise to applications of discrete curvature in node clustering and community detection. For downstream machine learning and data science applications, it is often beneficial to represent graph-structured data in a continuous space, which may be Euclidean or Non-Euclidean. We show that discrete curvature allows for characterizing the geometry of a suitable embedding space both locally and in the sense of global curvature bounds, which have implications for graph-based learning.